Expression of type LessEq

from the theory of proveit.physics.quantum.QPE

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import l
from proveit.numbers import Abs, Add, Exp, LessEq, Mult, Sum, four, frac, one, two
from proveit.physics.quantum.QPE import _diff_l_scaled_delta_floor, _neg_domain, _pos_domain, _rel_indexed_alpha

# build up the expression from sub-expressions
sub_expr1 = [l]
sub_expr2 = Exp(Abs(_rel_indexed_alpha), two)
sub_expr3 = Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _pos_domain)
expr = LessEq(Add(Sum(index_or_indices = sub_expr1, summand = sub_expr2, domain = _neg_domain), sub_expr3), Add(Sum(index_or_indices = sub_expr1, summand = frac(one, Mult(four, Exp(_diff_l_scaled_delta_floor, two))), domain = _neg_domain), sub_expr3))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right)\right) \leq \left(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right) + \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 70 operands: 5
4	Operation	operator: 70 operands: 6
5	ExprTuple	7, 9
6	ExprTuple	8, 9
7	Operation	operator: 12 operand: 14
8	Operation	operator: 12 operand: 15
9	Operation	operator: 12 operand: 16
10	ExprTuple	14
11	ExprTuple	15
12	Literal
13	ExprTuple	16
14	Lambda	parameter: 63 body: 17
15	Lambda	parameter: 63 body: 18
16	Lambda	parameter: 63 body: 20
17	Conditional	value: 23 condition: 22
18	Conditional	value: 21 condition: 22
19	ExprTuple	63
20	Conditional	value: 23 condition: 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 29 operands: 27
23	Operation	operator: 75 operands: 28
24	Operation	operator: 29 operands: 30
25	Literal
26	ExprTuple	84, 31
27	ExprTuple	63, 32
28	ExprTuple	33, 81
29	Literal
30	ExprTuple	63, 34
31	Operation	operator: 68 operands: 35
32	Operation	operator: 39 operands: 36
33	Operation	operator: 37 operand: 45
34	Operation	operator: 39 operands: 40
35	ExprTuple	41, 42
36	ExprTuple	43, 44
37	Literal
38	ExprTuple	45
39	Literal
40	ExprTuple	53, 61
41	Literal
42	Operation	operator: 75 operands: 46
43	Operation	operator: 70 operands: 47
44	Operation	operator: 79 operand: 53
45	Operation	operator: 49 operand: 54
46	ExprTuple	51, 81
47	ExprTuple	52, 84
48	ExprTuple	53
49	Literal
50	ExprTuple	54
51	Operation	operator: 70 operands: 55
52	Operation	operator: 79 operand: 61
53	Operation	operator: 70 operands: 57
54	Operation	operator: 58 operands: 59
55	ExprTuple	63, 60
56	ExprTuple	61
57	ExprTuple	62, 84
58	Literal
59	ExprTuple	83, 63
60	Operation	operator: 79 operand: 66
61	Operation	operator: 75 operands: 65
62	Variable
63	Variable
64	ExprTuple	66
65	ExprTuple	81, 67
66	Operation	operator: 68 operands: 69
67	Operation	operator: 70 operands: 71
68	Literal
69	ExprTuple	72, 73
70	Literal
71	ExprTuple	82, 74
72	Operation	operator: 75 operands: 76
73	Operation	operator: 77 operand: 83
74	Operation	operator: 79 operand: 84
75	Literal
76	ExprTuple	81, 82
77	Literal
78	ExprTuple	83
79	Literal
80	ExprTuple	84
81	Literal
82	Literal
83	Literal
84	Literal